AP Calculus BC Series and Sequences Practice Problems (2025 Guide)

Introduction: Why Series and Sequences Matter for AP Calculus BC

If you’re preparing for the AP Calculus BC exam, you already know that series and sequences make up a significant chunk of the test. These problems test your ability to analyze convergence, apply various tests, and work with Taylor and Maclaurin series.

Many students struggle with this unit because it requires not just algebra and calculus, but also logical reasoning about infinite processes. The good news? With the right practice problems and strategies, you can master sequences and series and secure major points on the exam.

In this guide, we’ll cover:

• The key topics in sequences and series for AP Calculus BC.
• The most important convergence tests (and how to know which to use).
• Step-by-step practice problems with solutions.
• Common mistakes to avoid.
• A set of additional practice questions.

For full study guides, past exam solutions, and practice sets, check out RevisionDojo’s AP Calculus BC resources.

Key Topics in Sequences and Series for AP Calculus BC

Before diving into practice problems, let’s outline what you need to know:

• Sequences: Limits of sequences, monotone convergence, boundedness.
• Infinite Series: Convergence vs. divergence, partial sums.
• Convergence Tests: Geometric series, nth-term test, p-series, ratio test, root test, alternating series test, integral test, direct/limit comparison tests.
• Power Series: Interval of convergence, radius of convergence.
• Taylor and Maclaurin Series: Expansions for common functions, error bounds.

These concepts appear not just in multiple-choice questions but also in free-response, often requiring explanation and justification.

Step-by-Step Practice Problem 1: Convergence of a Series

Problem: Determine whether the series

∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}

converges or diverges.

Solution:

1. Recognize this as a p-series with p = 2.
2. Rule: A p-series converges if p > 1.
3. Since p = 2 > 1, the series converges.

Final Answer: The series converges.

Step-by-Step Practice Problem 2: The Ratio Test

Problem: Does the series

∑n=1∞3nn!\sum_{n=1}^{\infty} \frac{3^n}{n!}

converge or diverge?

Solution:

1. Apply the Ratio Test: L=lim⁡n→∞∣an+1an∣L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
2. Compute: an+1an=3n+1/(n+1)!3n/n!=3n+1.\frac{a_{n+1}}{a_n} = \frac{3^{n+1}/(n+1)!}{3^n/n!} = \frac{3}{n+1}.
3. Take the limit: L=lim⁡n→∞3n+1=0.L = \lim_{n \to \infty} \frac{3}{n+1} = 0.
4. Since L < 1, the series converges absolutely.

Final Answer: The series converges by the Ratio Test.

Step-by-Step Practice Problem 3: Alternating Series

Problem: Does the series

∑n=1∞(−1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{n}

converge?

Solution:

1. This is the Alternating Harmonic Series.
2. Apply the Alternating Series Test:
   • Terms decrease in absolute value: 1/n decreases.
   • Limit of terms = 0.
3. Both conditions satisfied → series converges.

Final Answer: The series converges conditionally.

Step-by-Step Practice Problem 4: Interval of Convergence

Problem: Find the interval of convergence for the power series

∑n=0∞(x−2)nn⋅3n\sum_{n=0}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}

Solution:

1. Apply the Ratio Test: L=lim⁡n→∞∣(x−2)n+1/((n+1)3n+1)(x−2)n/(n3n)∣=lim⁡n→∞∣x−2∣3⋅nn+1.L = \lim_{n \to \infty} \left| \frac{(x-2)^{n+1}/((n+1)3^{n+1})}{(x-2)^n/(n3^n)} \right| = \lim_{n \to \infty} \frac{|x-2|}{3} \cdot \frac{n}{n+1}.
2. Simplify: L=∣x−2∣3.L = \frac{|x-2|}{3}.
3. Convergence when L < 1 → |x – 2| < 3.
4. Interval: (–1, 5). Must test endpoints:
   • At x = –1: series = ∑ (–3)^n/(n3^n) = ∑ (–1)^n/n → converges (alternating harmonic).
   • At x = 5: series = ∑ 3^n/(n3^n) = ∑ 1/n → diverges (harmonic series).

Final Answer: Interval of convergence = [–1, 5).

Step-by-Step Practice Problem 5: Maclaurin Series Expansion

Problem: Write the Maclaurin series for sin(x) up to the x⁵ term.

Solution:

1. Formula: sin(x) = x – x³/3! + x⁵/5! – …
2. Up to x⁵: sin(x) ≈ x – x³/6 + x⁵/120.

Final Answer:

sin⁡(x)≈x−x36+x5120\sin(x) \approx x - \frac{x^3}{6} + \frac{x^5}{120}

Common Mistakes in Sequences and Series

• Forgetting that nth-term test only proves divergence, not convergence.
• Mixing up absolute vs. conditional convergence.
• Forgetting to check endpoints in interval of convergence problems.
• Misusing Taylor expansions (not aligning with factorials correctly).
• Leaving justifications incomplete on FRQs (always cite the correct convergence test).

Pro Tips for Scoring on Sequences and Series

• Always write the name of the test you’re applying (Ratio Test, Root Test, etc.). The AP readers want explicit justification.
• When working with Taylor/Maclaurin series, memorize expansions for:
  • sin(x), cos(x), e^x, ln(1+x), 1/(1–x).
• Clearly separate steps when testing endpoints.
• Practice writing full sentences: “By the Ratio Test, since L < 1, the series converges absolutely.”
• Use practice to spot which convergence test applies to each series type.

Additional Practice Problems

1. Determine if the series ∑ (2^n / n³) converges or diverges.
2. Find the radius and interval of convergence for ∑ (x^n / n²).
3. Write the first four nonzero terms of the Maclaurin series for e^(–x²).
4. Does ∑ (–1)^n (n / (n²+1)) converge absolutely, converge conditionally, or diverge?
5. Evaluate ∑ (1 / 3^n) from n=1 to infinity.

(Step-by-step solutions are available in RevisionDojo’s AP Calculus BC series library.)

Frequently Asked Questions

1. What types of series and sequences appear on the AP Calculus BC exam?
You’ll see convergence tests, power series, and Taylor/Maclaurin expansions.

2. Do I need to memorize series for the AP exam?
Yes — especially the Maclaurin series for e^x, sin(x), cos(x), ln(1+x), and 1/(1–x).

3. What’s the most important convergence test?
The Ratio Test is the most commonly applied, especially for factorials and exponentials.

4. How can I avoid mistakes with intervals of convergence?
Always test endpoints separately and justify using the correct convergence test.

5. Where can I find AP-style practice series problems with solutions?
Check RevisionDojo’s AP Calculus BC resources for worked examples and exam-style problems.

Conclusion: Mastering Series and Sequences for AP Success

Series and sequences can feel overwhelming, but they follow clear patterns once you practice the main tests. By mastering p-series, ratio tests, alternating series, and Taylor expansions, you’ll be well-prepared for both multiple-choice and FRQs on the AP Calculus BC exam.

The best way to build confidence is through consistent practice. Work through a wide range of convergence problems, explain your reasoning clearly, and memorize the core Maclaurin series expansions. For structured study plans and expert walkthroughs, visit RevisionDojo and take your AP Calculus prep to the next level.